The Instability of the Hocking-Stewartson Pulse and its Geometric Phase in the Hopf Bundle

TL;DR

This paper introduces a novel numerical approach using the geometric phase in the Hopf bundle to accurately count and locate eigenvalues of the Hocking-Stewartson pulse in the complex Ginzburg-Landau equation, enhancing spectral analysis techniques.

Contribution

It redevelops Way's method for the Hocking-Stewartson pulse with a modified numerical shooting approach and presents new results on phase transition phenomena.

Findings

Successful eigenvalue counting using the geometric phase method

Enhanced numerical stability and accuracy in locating eigenvalues

New insights into phase transition behavior

Abstract

This work demonstrates an innovative numerical method for counting and locating eigenvalues with the Evans function. Utilizing the geometric phase in the Hopf bundle, the technique calculates the winding of the Evans function about a contour in the spectral plane, describing the eigenvalues enclosed by the contour for the Hocking-Stewartson pulse of the complex Ginzburg-Landau equation. Locating eigenvalues with the geometric phase in the Hopf bundle was proposed by Way, and proven by Grudzien, Bridges & Jones. Way demonstrated his proposed method for the Hocking-Stewartson pulse, and this manuscript redevelops this example as in the proof of the method, modifying his numerical shooting argument, and introduces new numerical results concerning the phase transition.